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Abstract

Using linear arithmetic operations, the floor operation, and indirect
addressing, we sequentially compute $GCD(x,y)$ for $0~<=~x,y~<=~N$, and
find two integers $-N~<=~a,b~<=~N$ such that $ax~+~by~=~GCD(x,y)$ with
operation complexity
0 left ( {log~N} over {log~log~N} right )
and space complexity $0(($log$~N) sup epsilon )$ for any constant
$0~<~epsilon~<~1$.
The intermediate numbers obtained during the execution of these algorithms
are all less than $max(x,y)$. In the second part of this paper (Part II.
Lower Bound), we prove that, using the given operations, this bound is
tight.
We also study the direct sum complexity of computing the GCD and also
the complexity of computing the GCD using tables of bounded size.

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